3.894 \(\int \frac{1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=493 \[ -\frac{b^{3/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^{3/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-3 a d)}{a c^2 e \sqrt{e x} (b c-a d)}-\frac{d}{c e \sqrt{e x} \sqrt{c-d x^2} (b c-a d)} \]

[Out]

-(d/(c*(b*c - a*d)*e*Sqrt[e*x]*Sqrt[c - d*x^2])) - ((2*b*c - 3*a*d)*Sqrt[c - d*x
^2])/(a*c^2*(b*c - a*d)*e*Sqrt[e*x]) - (d^(1/4)*(2*b*c - 3*a*d)*Sqrt[1 - (d*x^2)
/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*c^(5/4)*(b*
c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/4)*(2*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]
*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*c^(5/4)*(b*c -
a*d)*e^(3/2)*Sqrt[c - d*x^2]) - (b^(3/2)*c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi
[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqr
t[e])], -1])/(a^(3/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (b^(3/2)*c^
(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin
[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a^(3/2)*d^(1/4)*(b*c - a*d)*e^(3/
2)*Sqrt[c - d*x^2])

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Rubi [A]  time = 2.76648, antiderivative size = 493, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433 \[ -\frac{b^{3/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^{3/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a c^{5/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-3 a d)}{a c^2 e \sqrt{e x} (b c-a d)}-\frac{d}{c e \sqrt{e x} \sqrt{c-d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

-(d/(c*(b*c - a*d)*e*Sqrt[e*x]*Sqrt[c - d*x^2])) - ((2*b*c - 3*a*d)*Sqrt[c - d*x
^2])/(a*c^2*(b*c - a*d)*e*Sqrt[e*x]) - (d^(1/4)*(2*b*c - 3*a*d)*Sqrt[1 - (d*x^2)
/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*c^(5/4)*(b*
c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/4)*(2*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]
*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*c^(5/4)*(b*c -
a*d)*e^(3/2)*Sqrt[c - d*x^2]) - (b^(3/2)*c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi
[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqr
t[e])], -1])/(a^(3/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (b^(3/2)*c^
(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin
[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a^(3/2)*d^(1/4)*(b*c - a*d)*e^(3/
2)*Sqrt[c - d*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x)**(3/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.8487, size = 401, normalized size = 0.81 \[ \frac{x \left (\frac{49 c x^2 \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) (b c-a d) \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{3 \left (\frac{11 b c d x^4 (3 a d-2 b c) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{14 b c \left (c-d x^2\right )}{a}+7 d \left (3 d x^2-2 c\right )\right )}{a d-b c}\right )}{21 c^2 (e x)^{3/2} \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

(x*((49*c*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x^2*AppellF1[3/4, 1/2, 1, 7/4, (d*
x^2)/c, (b*x^2)/a])/((b*c - a*d)*(a - b*x^2)*(7*a*c*AppellF1[3/4, 1/2, 1, 7/4, (
d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b*c*AppellF1[7/4, 1/2, 2, 11/4, (d*x^2)/c, (b*x^
2)/a] + a*d*AppellF1[7/4, 3/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))) + (3*((14*b*c*(
c - d*x^2))/a + 7*d*(-2*c + 3*d*x^2) + (11*b*c*d*(-2*b*c + 3*a*d)*x^4*AppellF1[7
/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])/((a - b*x^2)*(11*a*c*AppellF1[7/4, 1/2,
1, 11/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b*c*AppellF1[11/4, 1/2, 2, 15/4, (d*x
^2)/c, (b*x^2)/a] + a*d*AppellF1[11/4, 3/2, 1, 15/4, (d*x^2)/c, (b*x^2)/a])))))/
(-(b*c) + a*d)))/(21*c^2*(e*x)^(3/2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.043, size = 1058, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x)^(3/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)

[Out]

1/2*(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1
/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*b^2*c^3-(c*d)^(1/2)
*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*b*c^2+((d*x
+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^
(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*b^2*c^3+(c*d)^(1/2)*((d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*
d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*b*c^2+6*((d*x+(c*d)^(
1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c
*d)^(1/2))^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^
2*c*d^2-10*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*
d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2
))^(1/2),1/2*2^(1/2))*a*b*c^2*d+4*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticE(((d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^3-3*((d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(
1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c*d^2+5*((
d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*
2^(1/2))*a*b*c^2*d-2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^
(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^3+6*x^2*a^2*d^3-10*x^2*a*b*c*d^2+4*x^2*b^2
*c^2*d-4*a^2*c*d^2+8*a*b*c^2*d-4*b^2*c^3)*b*d*(-d*x^2+c)^(1/2)/c^2/((c*d)^(1/2)*
b-(a*b)^(1/2)*d)/((a*b)^(1/2)*d+(c*d)^(1/2)*b)/(a*d-b*c)/a/(d*x^2-c)/e/(e*x)^(1/
2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(3/2)),x, algorithm="maxima")

[Out]

-integrate(1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(3/2)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(3/2)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x)**(3/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(3/2)),x, algorithm="giac")

[Out]

integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(3/2)), x)